Sample Spaces and Events

Probability starts by naming possible outcomes and the events made from them.

Probability begins before any formula. First we must name what can happen. A coin can land heads or tails. A die can show one of six faces. A selected student may be in one group or another. These possible results form the sample space. Questions such as “rolling an even number” or “getting heads” are events: subsets of that sample space.

Learning objectives

After this lesson you should be able to:

  • define random experiment, outcome, sample space, and event;
  • write events as subsets of a sample space;
  • compute simple probabilities by counting favorable outcomes;
  • use complements, unions, and intersections of events;
  • recognize when outcomes are not equally likely;
  • check finite event calculations with Python, edumath, and SymPy.

Random experiments, outcomes, and sample spaces

A random experiment is a process whose result is uncertain before it happens. An outcome is one possible result. The sample space, often written S, is the set of all possible outcomes.

For one coin flip:

S = {H, T}

For one fair die roll:

S = {1, 2, 3, 4, 5, 6}

The sample space must match the question. If the experiment is “flip two coins,” then {H, T} is too small. A better sample space is:

S = {HH, HT, TH, TT}

Events as subsets

An event is a set of outcomes inside the sample space. For a die roll,

S = {1, 2, 3, 4, 5, 6}

The event “roll an even number” is

E = {2, 4, 6}

The event “roll greater than 4” is

G = {5, 6}

An event can have one outcome, many outcomes, all outcomes, or no outcomes.

Equally likely outcomes

When all outcomes in S are equally likely, probability is counting:

P(event) = number of favorable outcomes / number of total outcomes

For a fair die, the probability of rolling an even number is

P(E) = 3 / 6 = 1 / 2

because E = {2,4,6} has 3 favorable outcomes out of 6 total outcomes.

WarningEqual likelihood matters

Counting outcomes works only when the outcomes are equally likely. If a coin is weighted, {H, T} is still the sample space, but heads and tails may not each have probability 1/2.

Complements, unions, and intersections

Events use the same language as sets.

  • The complement of A means “not A.”
  • The union A or B means outcomes in A, in B, or in both.
  • The intersection A and B means outcomes in both A and B.

For a fair die:

E = {2, 4, 6}      even
G = {5, 6}         greater than 4

Then:

not E      = {1, 3, 5}
E or G     = {2, 4, 5, 6}
E and G    = {6}

Probability rules you should know early

For any event A:

0 <= P(A) <= 1
P(S) = 1
P(not A) = 1 - P(A)

If A and B cannot happen at the same time, then

P(A or B) = P(A) + P(B)

If they can overlap, subtract the overlap once:

P(A or B) = P(A) + P(B) - P(A and B)

Worked example 1: one die roll

Roll a fair die. Find the probability of rolling an even number or a number greater than 4.

  1. List the sample space:

    S = {1, 2, 3, 4, 5, 6}

  2. Name the events:

    E = {2, 4, 6}
G = {5, 6}

  3. Combine using “or”:

    E or G = {2, 4, 5, 6}

  4. Count favorable outcomes:

    P(E or G) = 4/6 = 2/3

Worked example 2: complement strategy

Suppose you flip three fair coins. What is the probability of getting at least one head?

Direct counting is possible, but the complement is easier.

  1. “At least one head” means one, two, or three heads.

  2. The complement is “no heads,” which is only TTT.

  3. There are 2^3 = 8 equally likely outcomes.

  4. So

    P(no heads) = 1/8
P(at least one head) = 1 - 1/8 = 7/8

Common pitfalls

  • Forgetting to define the sample space.
  • Treating outcomes as equally likely when they are not.
  • Confusing one outcome, such as 6, with an event, such as {6}.
  • Counting an overlap twice when using “or.”
  • Forgetting that complements depend on the sample space.

Practice exercises

  1. What is the sample space for one fair coin flip?
  2. For a fair die, write the event “roll an odd number.”
  3. For a fair die, compute P(roll at least 5).
  4. For a fair die, compute P(not even).
  5. For two fair coins, list the sample space and the event “exactly one head.”
  1. {H, T}.
  2. {1, 3, 5}.
  3. {5, 6} has 2 outcomes out of 6, so the probability is 2/6 = 1/3.
  4. Not even means {1, 3, 5}, so the probability is 3/6 = 1/2.
  5. S = {HH, HT, TH, TT}. Exactly one head is {HT, TH}.

Subtopic guided practice and checkpoints

Probability problems are easiest when you translate words into sets before doing arithmetic.

Lab 1: list the sample space

Guess first. If you flip two coins, are there 2 outcomes or 4 outcomes?

Guided exercise.

Write every ordered result:

HH, HT, TH, TT

There are 4 outcomes because the first flip and second flip are separate parts of the experiment.

Checkpoint. List the sample space for tossing one coin and rolling one die. How many outcomes are there?

Lab 2: turn words into event subsets

Guess first. For a die roll, is “less than 3” the event {1,2} or {3,4,5,6}?

Guided exercise.

Start with S = {1,2,3,4,5,6}. The words “less than 3” select outcomes 1 and 2:

A = {1, 2}

Checkpoint. For a die roll, write the event “odd and greater than 2.”

Lab 3: compute probability by counting

Guess first. If an event has 3 favorable outcomes out of 8 equally likely outcomes, is the probability 3/8 or 8/3?

Guided exercise.

Probability is favorable divided by total:

P(event) = 3 / 8

The denominator is the size of the sample space.

Checkpoint. A bag contains 4 red, 3 blue, and 3 green marbles. If one marble is chosen at random, what is P(blue)?

Lab 4: use complements

Guess first. Is “not rolling a 6” easier to count than “rolling 1, 2, 3, 4, or 5”?

Guided exercise.

For a fair die:

P(6) = 1/6
P(not 6) = 1 - 1/6 = 5/6

The complement can save time when the event has many pieces.

Checkpoint. In three coin flips, use a complement to find the probability of getting at least one tail.

Lab 5: check equal likelihood

Guess first. In a weighted coin, are {H} and {T} automatically equally likely?

Guided exercise.

The sample space is still {H,T}, but the probabilities might be different, such as:

P(H) = 0.7
P(T) = 0.3

Counting outcomes alone would incorrectly give 1/2 for each.

Checkpoint. Give one example of a sample space where outcomes might not be equally likely.

  1. {H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6}; there are 12 outcomes.
  2. Odd and greater than 2 is {3,5}.
  3. There are 10 marbles total and 3 blue marbles, so P(blue)=3/10.
  4. The complement of at least one tail is no tails, or HHH. Probability is 1 - 1/8 = 7/8.
  5. Examples include weighted coins, biased dice, weather outcomes, and survey responses.

Sample-space guessing game

Use this checkpoint to practice outcomes, events, complements, and simple probabilities.

Using this lesson with edumath and SymPy

Use Python sets to model finite sample spaces.

sample_space = {1, 2, 3, 4, 5, 6}
even = {2, 4, 6}
greater_than_four = {5, 6}

not_even = sample_space - even
even_or_greater_than_four = even | greater_than_four
even_and_greater_than_four = even & greater_than_four

not_even, even_or_greater_than_four, even_and_greater_than_four
({1, 3, 5}, {2, 4, 5, 6}, {6})

Use edumath to compute probabilities from finite equally likely outcomes.

from edumath.probability import complement, probability

probability(even, sample_space), complement(even, sample_space)
(0.5, {1, 3, 5})

Use SymPy for exact fraction arithmetic.

import sympy as sp

sp.Rational(len(even | greater_than_four), len(sample_space))

\(\displaystyle \frac{2}{3}\)